组合数学第三次作业

组合数学第三次作业
1.
(1) we choose $x_i$ denotes the $i^{th}$ coin,and $T_i=\{x_i\}$for i in {1,..n}
and according to an arbitrary subset $T_i$ can be uniquely determined by the cardinalities$|S_i\cap T|, 1\le i\le m.$
so we can weight every coin.
(2)

$$m\ge \frac{n}{\log_2(n+1)}$$
is equvient with $$(n+1)*m>2^n$$
the number of $T_i$ is $2^n$, and the number of $S_i$ is $(n+1)*m$
and it’s easy to see that $||S_i||>||T_i||$

2.
(1)
let $T$ is the set of vertex which for every vetex $v_i$ in $V$, we have an independent probability $p$ to put $v_i$ into T
and $W$ is the set of vertex $v_j, v_j\notin T,$and every neighbour of $v_j \notin T$
So,

$$E[T]=np$$
$$E[W]=n(1-p)^{d+1}$$
the dominating set $$D=T \cup W$$
$$D<np+n(1-p)^{d+1}=H(p)$$
when$H'(p)=0,p=\frac{ln(d+1)}{d+1}$
and we have $$D<\frac{n(1+\ln(d+1))}{d+1}$$

3.

4.let $A_{ij}$represents the event that$v_i,v_j$ are adjacent,and $v_i,v_j$are chosen in the independent set.

$$P_{A_{ij}}=\frac{1}{k^2}$$
and d=2
According to Lovász local lemma,we have to prove
$$ep(d+1)<1$$
$$e*\frac{1}{k^2}*2<1$$
for any k>3, it holds.

5.
let $A_e$ denotes the bed event that $e$ is a monochromatic hyper edge .

$$P_{A_e}=2^{1-k}$$
the maximum degree of the dependency graph for the events ${A_1,A_2,...,A_n}$ is $d$,
$$d=2k$$
According to Lovász local lemma
we have to prove

$$ep(d+1)<1$$ for k>10
$$ep(d+1)<e*2^{1-k}*2k<1$$